Addition and Subtraction — Practice Quiz

A Trigonometry cheat sheet for Addition and Subtraction — every key formula with its symbols defined — plus a medium-level practice quiz to test recall.

Formulas & key concepts

Sine Sum

$$\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$

Sine Difference

$$\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$

Cosine Sum

$$\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$

Cosine Difference

$$\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$

Tangent Sum

$$\tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$

Tangent Difference

$$\tan(\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$$

Practice quiz

  1. What is the expansion of $\sin(A+B)$?

    • $\sin A \cos B + \cos A \sin B$
    • $\sin A \cos B - \cos A \sin B$
    • $\cos A \cos B + \sin A \sin B$
    • $\cos A \cos B - \sin A \sin B$

    Answer: $\sin A \cos B + \cos A \sin B$

  2. Which of the following is equivalent to $\cos(x-y)$?

    • $\cos x \cos y - \sin x \sin y$
    • $\cos x \cos y + \sin x \sin y$
    • $\sin x \cos y + \cos x \sin y$
    • $\sin x \cos y - \cos x \sin y$

    Answer: $\cos x \cos y + \sin x \sin y$

  3. The formula for $\tan(\alpha + \beta)$ is:

    • $\frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$
    • $\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$
    • $\frac{\tan \alpha + \tan \beta}{1 + \tan \alpha \tan \beta}$
    • $\frac{\tan \alpha - \tan \beta}{1 - \tan \alpha \tan \beta}$

    Answer: $\frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$

  4. Evaluate the exact value of $\sin(75^\circ)$.

    • $\frac{\sqrt{6} + \sqrt{2}}{4}$
    • $\frac{\sqrt{6} - \sqrt{2}}{4}$
    • $\frac{\sqrt{3} + 1}{2}$
    • $\frac{\sqrt{3} - 1}{2}$

    Answer: $\frac{\sqrt{6} + \sqrt{2}}{4}$

  5. Simplify the expression $\cos(60^\circ) \cos(30^\circ) - \sin(60^\circ) \sin(30^\circ)$.

    • $\cos(30^\circ)$
    • $\sin(30^\circ)$
    • $\cos(90^\circ)$
    • $\sin(90^\circ)$

    Answer: $\cos(90^\circ)$

  6. The expression $\sin(5x) \cos(2x) - \cos(5x) \sin(2x)$ simplifies to:

    • $\sin(7x)$
    • $\sin(3x)$
    • $\cos(7x)$
    • $\cos(3x)$

    Answer: $\sin(3x)$

  7. What is the exact value of $\tan(15^\circ)$?

    • $2 + \sqrt{3}$
    • $2 - \sqrt{3}$
    • $\frac{1 + \sqrt{3}}{1 - \sqrt{3}}$
    • $\frac{1 - \sqrt{3}}{1 + \sqrt{3}}$

    Answer: $2 - \sqrt{3}$

  8. Given that $\sin A = \frac{3}{5}$ (A in Q1) and $\cos B = \frac{5}{13}$ (B in Q1), find $\cos(A+B)$.

    • $\frac{16}{65}$
    • $-\frac{16}{65}$
    • $\frac{56}{65}$
    • $-\frac{56}{65}$

    Answer: $-\frac{16}{65}$

  9. Which of the following trigonometric identities is incorrect?

    • $\sin(x-y) = \sin x \cos y - \cos x \sin y$
    • $\cos(x+y) = \cos x \cos y - \sin x \sin y$
    • $\tan(x+y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$
    • $\cos(x-y) = \cos x \cos y - \sin x \sin y$

    Answer: $\cos(x-y) = \cos x \cos y - \sin x \sin y$

  10. Simplify $\frac{\sin(A+B)}{\cos A \cos B}$.

    • $\tan A + \tan B$
    • $\tan A - \tan B$
    • $\cot A + \cot B$
    • $\cot A - \cot B$

    Answer: $\tan A + \tan B$

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